Dynamic Foraging Patterns in Antarctic Humpbacks
Ben Weinstein
2017-05-23 17:55:50
## Source: local data frame [12 x 2]
##
## Animal max(timestamp, na.rm = T)
## (int) (time)
## 1 112699 2012-06-17 03:57:31
## 2 121207 2013-05-09 18:49:00
## 3 121208 2013-02-18 07:52:00
## 4 121210 2013-05-05 07:44:00
## 5 123224 2013-05-24 12:13:00
## 6 123232 2013-09-28 07:28:00
## 7 123236 2013-03-18 11:26:00
## 8 131127 2016-07-15 07:58:36
## 9 131130 2016-04-30 00:30:06
## 10 131132 2016-05-10 19:44:39
## 11 131133 2016-07-05 20:26:44
## 12 131136 2016-06-30 18:49:35
0.1 By Month
1 Correlated random walk
Process Model
\[ d_{t} \sim T*d_{t-1} + Normal(0,\Sigma)\] \[ x_t = x_{t-1} + d_{t} \]
1.1 Parameters
For each individual:
\[\theta = \text{Mean turning angle}\] \[\gamma = \text{Move persistence} \]
For both behaviors process variance is: \[ \sigma_{latitude} = 0.1\] \[ \sigma_{longitude} = 0.1\]
1.2 Behavioral States
\[ \text{For each individual i}\] \[ Behavior_1 = \text{traveling}\] \[ Behavior_2 = \text{foraging}\]
\[ \alpha_{i,1,1} = \text{Probability of remaining traveling when traveling}\] \[\alpha_{i,2,1} = \text{Probability of switching from Foraging to traveling}\]
\[\begin{matrix} \alpha_{i,1,1} & 1-\alpha_{i,1,1} \\ \alpha_{i,2,1} & 1-\alpha_{i,2,1} \\ \end{matrix}\]
With the probability of switching states:
\[logit(\phi_{traveling}) = \alpha_{Behavior_{t-1}}\]
\[\phi_{foraging} = 1 - \phi_{traveling} \]
1.3 Continious tracks
The transmitter will often go dark for 10 to 12 hours, due to weather, right in the middle of an otherwise good track. The model requires regular intervals to estimate the turning angles and temporal autocorrelation. As a track hits one of these walls, call it the end of a track, and begin a new track once the weather improves. We can remove any micro-tracks that are less than three days. Specify a duration, calculate the number of tracks and the number of removed points. Iteratively.
1.3.1 After filitering
How did the filter change the extent of tracks?
sink(“Bayesian/Multi_RW.jags”) cat(" model{
#Constants
pi <- 3.141592653589
#for each if 6 argos class observation error
for(x in 1:6){
##argos observation error##
argos_prec[x,1:2,1:2] <- argos_cov[x,,]
#Constructing the covariance matrix
argos_cov[x,1,1] <- argos_sigma[x]
argos_cov[x,1,2] <- 0
argos_cov[x,2,1] <- 0
argos_cov[x,2,2] <- argos_alpha[x]
}
for(i in 1:ind){
for(g in 1:tracks[i]){
## Priors for first true location
#for lat long
y[i,g,1,1:2] ~ dmnorm(argos[i,g,1,1,1:2],argos_prec[1,1:2,1:2])
#First movement - random walk.
y[i,g,2,1:2] ~ dmnorm(y[i,g,1,1:2],iSigma)
###First Behavioral State###
state[i,g,1] ~ dcat(lambda[]) ## assign state for first obs
#Process Model for movement
for(t in 2:(steps[i,g]-1)){
#Behavioral State at time T
phi[i,g,t,1] <- alpha[state[i,g,t-1],Month[i,g,t]]
phi[i,g,t,2] <- 1-phi[i,g,t,1]
state[i,g,t] ~ dcat(phi[i,g,t,])
#Turning covariate
#Transition Matrix for turning angles
T[i,g,t,1,1] <- cos(theta[state[i,g,t]])
T[i,g,t,1,2] <- (-sin(theta[state[i,g,t]]))
T[i,g,t,2,1] <- sin(theta[state[i,g,t]])
T[i,g,t,2,2] <- cos(theta[state[i,g,t]])
#Correlation in movement change
d[i,g,t,1:2] <- y[i,g,t,] + gamma[state[i,g,t],Month[i,g,t]] * T[i,g,t,,] %*% (y[i,g,t,1:2] - y[i,g,t-1,1:2])
#Gaussian Displacement
y[i,g,t+1,1:2] ~ dmnorm(d[i,g,t,1:2],iSigma)
}
#Final behavior state
phi[i,g,steps[i,g],1] <- alpha[state[i,g,steps[i,g]-1],Month[i,g,steps[i,g]-1]]
phi[i,g,steps[i,g],2] <- 1-phi[i,g,steps[i,g],1]
state[i,g,steps[i,g]] ~ dcat(phi[i,g,steps[i,g],])
## Measurement equation - irregular observations
# loops over regular time intervals (t)
for(t in 2:steps[i,g]){
# loops over observed locations within interval t
for(u in 1:idx[i,g,t]){
zhat[i,g,t,u,1:2] <- (1-j[i,g,t,u]) * y[i,g,t-1,1:2] + j[i,g,t,u] * y[i,g,t,1:2]
#for each lat and long
#argos error
argos[i,g,t,u,1:2] ~ dmnorm(zhat[i,g,t,u,1:2],argos_prec[argos_class[i,g,t,u],1:2,1:2])
}
}
}
}
###Priors###
#Process Variance
iSigma ~ dwish(R,2)
Sigma <- inverse(iSigma)
##Mean Angle
tmp[1] ~ dbeta(10, 10)
tmp[2] ~ dbeta(10, 10)
# prior for theta in 'traveling state'
theta[1] <- (2 * tmp[1] - 1) * pi
# prior for theta in 'foraging state'
theta[2] <- (tmp[2] * pi * 2)
##Move persistance
# prior for gamma (autocorrelation parameter) in state 1
#for each month
for (m in 1:Months){
#Intercepts
logit(alpha[1,m]) <- alpha_mu_invlogit[1,m]
alpha_mu_invlogit[1,m] ~ dnorm(alpha_mu[1],alpha_tau[1])
logit(alpha[2,m]) <- alpha_mu_invlogit[2,m]
alpha_mu_invlogit[2,m] ~ dnorm(alpha_mu[2],alpha_tau[2])
gamma[1,m] ~ dnorm(gamma_mu,gamma_tau) ## gamma for state 1
dev[m] ~ dbeta(1,1) ## a random deviate to ensure that gamma[1] > gamma[2]
gamma[2,m] <- gamma[1,m] * dev[m]
}
##Behavioral States
#Hierarchical structure across months
#Switching among states in inv.logit space
alpha_mu[1] ~ dnorm(0,0.386)
alpha_mu[2] ~ dnorm(0,0.386)
#Variance in state change per month
alpha_tau[1] ~ dunif(0,5)
alpha_tau[2] ~ dunif(0,5)
#Probability of behavior switching
lambda[1] ~ dbeta(1,1)
lambda[2] <- 1 - lambda[1]
#spatial autocorrelation priors, we know the true state is high autocorrelation
gamma_mu~dnorm(0.8,100)
#reasonable variance keeps the chains from wandering.
gamma_tau ~ dunif(100,500)
##Argos priors##
#longitudinal argos precision, from Jonsen 2005, 2016, represented as precision not sd
#by argos class
argos_sigma[1] <- 11.9016
argos_sigma[2] <- 10.2775
argos_sigma[3] <- 1.228984
argos_sigma[4] <- 2.162593
argos_sigma[5] <- 3.885832
argos_sigma[6] <- 0.0565539
#latitidunal argos precision, from Jonsen 2005, 2016
argos_alpha[1] <- 67.12537
argos_alpha[2] <- 14.73474
argos_alpha[3] <- 4.718973
argos_alpha[4] <- 0.3872023
argos_alpha[5] <- 3.836444
argos_alpha[6] <- 0.1081156
}"
,fill=TRUE)sink()
## user system elapsed
## 337.338 14.828 97254.0861.4 Chains
## used (Mb) gc trigger (Mb) max used (Mb)
## Ncells 1490299 79.6 3205452 171.2 3205452 171.2
## Vcells 1714633394 13081.7 4032004763 30761.8 3457122303 26375.8## used (Mb) gc trigger (Mb) max used (Mb)
## Ncells 1330370 71.1 3205452 171.2 3205452 171.2
## Vcells 146016614 1114.1 3225603810 24609.5 3457122303 26375.8
1.5 Change in autocorrelation over time
2 Change in transition probabilities over time
2.1 Parameter Summary
## parameter par mean lower upper
## 1 alpha alpha[1,1] 0.83802019 0.761248126 0.90612053
## 2 alpha alpha[2,1] 0.14317890 0.084073800 0.21495309
## 3 alpha alpha[1,2] 0.87296859 0.807546737 0.92467307
## 4 alpha alpha[2,2] 0.12249091 0.071840545 0.18916900
## 5 alpha alpha[1,3] 0.86305844 0.786676613 0.92539809
## 6 alpha alpha[2,3] 0.08023776 0.045901482 0.12386840
## 7 alpha alpha[1,4] 0.81645308 0.668781346 0.93813263
## 8 alpha alpha[2,4] 0.11736292 0.047220365 0.20650919
## 9 alpha alpha[1,5] 0.78277762 0.656528137 0.88449402
## 10 alpha alpha[2,5] 0.07049474 0.037936518 0.11173555
## 11 alpha alpha[1,6] 0.76821363 0.619891322 0.88894542
## 12 alpha alpha[2,6] 0.10854330 0.045065706 0.19556304
## 13 alpha_mu alpha_mu[1] 1.58447391 1.007321285 2.15959609
## 14 alpha_mu alpha_mu[2] -2.16539180 -2.694175247 -1.64174412
## 15 alpha_tau alpha_tau[1] 3.34698452 1.283325269 4.85325825
## 16 alpha_tau alpha_tau[2] 3.33273827 1.274820878 4.86538508
## 17 gamma gamma[1,1] 0.91296259 0.862549491 0.96496316
## 18 gamma gamma[2,1] 0.17617872 0.037863949 0.32390327
## 19 gamma gamma[1,2] 0.83134324 0.780860273 0.88737858
## 20 gamma gamma[2,2] 0.08535385 0.006840522 0.20630532
## 21 gamma gamma[1,3] 0.80865222 0.735332620 0.88794686
## 22 gamma gamma[2,3] 0.07734333 0.005983412 0.19668083
## 23 gamma gamma[1,4] 0.86514312 0.754764822 0.99033609
## 24 gamma gamma[2,4] 0.10780413 0.006218684 0.29597768
## 25 gamma gamma[1,5] 0.81909457 0.716411572 0.92365736
## 26 gamma gamma[2,5] 0.38637429 0.210061579 0.55143390
## 27 gamma gamma[1,6] 0.86634345 0.755787930 0.98500252
## 28 gamma gamma[2,6] 0.38447246 0.110143467 0.65417152
## 29 gamma_mu gamma_mu 0.84739652 0.783185851 0.91265592
## 30 gamma_tau gamma_tau 298.64687846 122.076514596 477.39130746
## 31 theta theta[1] 0.02440948 0.007619716 0.04175364
## 32 theta theta[2] 3.07400912 2.855138705 3.29013549
3 Behavioral Prediction
3.1 Spatial Prediction
3.2 By individual
3.3 Autocorrelation in behavior
3.3.1 As single timeline
4 Simulated tracks
4.1 Behavioral description
4.1.1 Proportion of states by month
4.1.2 Distance between bouts
5 Behavior duration
## Source: local data frame [166 x 7]
## Groups: Animal, Track, Bout, phistate [166]
##
## Animal Track Bout phistate MonthF Days Month
## (dbl) (dbl) (int) (fctr) (dbl) (dbl) (fctr)
## 1 1 1 1 Traveling 1 0.6059028 January
## 2 1 2 1 Traveling 1 0.4953819 January
## 3 1 3 1 Traveling 1 0.6483912 January
## 4 1 4 1 Traveling 1 1.5586921 January
## 5 1 5 1 Traveling 1 0.6500116 January
## 6 1 7 1 Traveling 2 0.9866435 February
## 7 1 8 1 Traveling 2 0.5460301 February
## 8 1 10 1 Traveling 2 0.9215972 February
## 9 1 11 1 Traveling 2 2.6218171 February
## 10 1 12 1 Traveling 2 0.6120602 February
## .. ... ... ... ... ... ... ...## Source: local data frame [177 x 7]
## Groups: Animal, Track, Bout, phistate [177]
##
## Animal Track Bout phistate MonthF Days Month
## (dbl) (dbl) (int) (fctr) (dbl) (dbl) (fctr)
## 1 1 6 1 Area-restricted Search 1 2.6031250 January
## 2 1 9 1 Area-restricted Search 2 3.3496065 February
## 3 2 1 2 Area-restricted Search 2 0.1564352 February
## 4 2 1 4 Area-restricted Search 2 0.1554977 February
## 5 2 2 2 Area-restricted Search 3 11.1066551 March
## 6 2 2 4 Area-restricted Search 3 0.9433218 March
## 7 2 2 6 Area-restricted Search 4 1.9837269 April
## 8 2 2 8 Area-restricted Search 4 3.9760880 April
## 9 2 2 10 Area-restricted Search 4 0.1050231 April
## 10 2 2 12 Area-restricted Search 4 0.9947569 April
## .. ... ... ... ... ... ... ...## Source: local data frame [343 x 7]
## Groups: Animal, Track, Bout, phistate [343]
##
## Animal Track Bout phistate MonthF Days Month
## (dbl) (dbl) (int) (fctr) (dbl) (dbl) (fctr)
## 1 1 1 1 Traveling 1 0.6059028 January
## 2 1 2 1 Traveling 1 0.4953819 January
## 3 1 3 1 Traveling 1 0.6483912 January
## 4 1 4 1 Traveling 1 1.5586921 January
## 5 1 5 1 Traveling 1 0.6500116 January
## 6 1 6 1 Area-restricted Search 1 2.6031250 January
## 7 1 7 1 Traveling 2 0.9866435 February
## 8 1 8 1 Traveling 2 0.5460301 February
## 9 1 9 1 Area-restricted Search 2 3.3496065 February
## 10 1 10 1 Traveling 2 0.9215972 February
## .. ... ... ... ... ... ... ...
5.1 Proportion of time allocation
## Month Traveling Area-restricted Search PropF TotalTime
## 1 January 121.93544 142.51822 0.5389157 264.45366
## 2 February 222.33936 331.34093 0.5984337 553.68029
## 3 March 110.77988 236.50453 0.6810111 347.28441
## 4 April 44.79596 150.25373 0.7703356 195.04969
## 5 May 24.35963 82.92464 0.7729431 107.28427
## 6 June 17.38661 44.18263 0.7176088 61.569245.2 Number of bouts
6 Time spent in grid cell
6.1 All years
6.2 Add in sea ice
